The document discusses the Cook-Toom algorithm for fast convolution. It reduces the number of multiplications in a linear convolution from NL to L+N-1 by using Lagrange interpolation. It expresses the convolution as a polynomial multiplication problem and evaluates the polynomials at distinct points, allowing the convolution to be computed using L+N-1 multiplications and additional additions by decomposing the convolution matrix. An example shows applying the algorithm to a 2x2 convolution, reducing multiplications from 4 to 3.
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdfpetercoiffeur18
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. It provides examples of dynamic programming approaches to computing Fibonacci numbers, binomial coefficients, the knapsack problem, and other problems. It also discusses greedy algorithms and provides examples of their application to problems like the change-making problem, minimum spanning trees, and single-source shortest paths.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
This document introduces simulation using MATLAB. It discusses how to generate random numbers from both discrete and continuous probability distributions using MATLAB commands or by converting uniformly distributed random numbers. For discrete distributions, the document shows how to generate random variables from the Bernoulli, binomial, Poisson, and geometric distributions. For continuous distributions, it explains the inverse transform method to generate random variables from exponential, gamma, normal, and other distributions by transforming uniformly distributed random numbers. The document also provides examples of MATLAB code to simulate coin tosses, generate random numbers from various distributions, and apply the Box-Muller transform to generate normal random variables. It concludes by reviewing useful MATLAB commands for commonly used discrete and continuous probability distributions.
EEE436 Lecture Slide 3.ppt digital commission codingKeepSmile24
The document provides lecture notes on error detection and correction using linear block codes like BCH and Reed-Solomon codes. It discusses syndrome decoding using parity check matrices to detect errors in received codewords. It also describes how to determine the generator polynomial for BCH codes based on the elements of the corresponding Galois field. Non-binary Reed-Solomon codes are also introduced which operate on symbols rather than individual bits.
This document introduces a generalized method for constructing sub-quadratic complexity multipliers for finite fields of characteristic 2. It begins by reintroducing the Winograd short convolution algorithm in the context of polynomial multiplication. It then presents a recursive construction technique that extends any d-point multiplier into an n=dk-point multiplier with sub-quadratic area and logarithmic delay complexity. Several new constructions are obtained using this technique, one of which is identical to the Karatsuba multiplier. The techniques aim to develop bit-parallel multipliers with better time and/or space complexity than the traditional quadratic complexity approaches.
The document describes the syllabus for a course on design analysis and algorithms. It covers topics like asymptotic notations, time and space complexities, sorting algorithms, greedy methods, dynamic programming, backtracking, and NP-complete problems. It also provides examples of algorithms like computing greatest common divisor, Sieve of Eratosthenes for primes, and discusses pseudocode conventions. Recursive algorithms and examples like Towers of Hanoi and permutation generation are explained. Finally, it outlines the steps for designing algorithms like understanding the problem, choosing appropriate data structures and computational devices.
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms based on their time complexity, space complexity, and correctness. It provides examples of analyzing simple algorithms and calculating their complexity based on the number of elementary operations.
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdfpetercoiffeur18
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. It provides examples of dynamic programming approaches to computing Fibonacci numbers, binomial coefficients, the knapsack problem, and other problems. It also discusses greedy algorithms and provides examples of their application to problems like the change-making problem, minimum spanning trees, and single-source shortest paths.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
This document introduces simulation using MATLAB. It discusses how to generate random numbers from both discrete and continuous probability distributions using MATLAB commands or by converting uniformly distributed random numbers. For discrete distributions, the document shows how to generate random variables from the Bernoulli, binomial, Poisson, and geometric distributions. For continuous distributions, it explains the inverse transform method to generate random variables from exponential, gamma, normal, and other distributions by transforming uniformly distributed random numbers. The document also provides examples of MATLAB code to simulate coin tosses, generate random numbers from various distributions, and apply the Box-Muller transform to generate normal random variables. It concludes by reviewing useful MATLAB commands for commonly used discrete and continuous probability distributions.
EEE436 Lecture Slide 3.ppt digital commission codingKeepSmile24
The document provides lecture notes on error detection and correction using linear block codes like BCH and Reed-Solomon codes. It discusses syndrome decoding using parity check matrices to detect errors in received codewords. It also describes how to determine the generator polynomial for BCH codes based on the elements of the corresponding Galois field. Non-binary Reed-Solomon codes are also introduced which operate on symbols rather than individual bits.
This document introduces a generalized method for constructing sub-quadratic complexity multipliers for finite fields of characteristic 2. It begins by reintroducing the Winograd short convolution algorithm in the context of polynomial multiplication. It then presents a recursive construction technique that extends any d-point multiplier into an n=dk-point multiplier with sub-quadratic area and logarithmic delay complexity. Several new constructions are obtained using this technique, one of which is identical to the Karatsuba multiplier. The techniques aim to develop bit-parallel multipliers with better time and/or space complexity than the traditional quadratic complexity approaches.
The document describes the syllabus for a course on design analysis and algorithms. It covers topics like asymptotic notations, time and space complexities, sorting algorithms, greedy methods, dynamic programming, backtracking, and NP-complete problems. It also provides examples of algorithms like computing greatest common divisor, Sieve of Eratosthenes for primes, and discusses pseudocode conventions. Recursive algorithms and examples like Towers of Hanoi and permutation generation are explained. Finally, it outlines the steps for designing algorithms like understanding the problem, choosing appropriate data structures and computational devices.
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms based on their time complexity, space complexity, and correctness. It provides examples of analyzing simple algorithms and calculating their complexity based on the number of elementary operations.
This document discusses methods for performing sparse time-frequency representation (STFR) on signals in 2 dimensions. STFR decomposes signals into intrinsic mode functions (IMFs) by finding the sparsest representation of the signal over a redundant dictionary. The 1D version has been implemented successfully, but extending to 2D is challenging due to the need to update in two directions simultaneously. Several attempted algorithms are described, including applying the 1D algorithm to slices of the 2D signal in different directions and averaging the results. The most recent attempt uses bi-directional slicing to overcome issues with previous global approaches.
Introducción al Análisis y diseño de algoritmosluzenith_g
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms to determine their time and space complexity, and how this involves determining how the resources required grow with the size of the problem. It provides examples of analyzing simple algorithms and determining whether they have linear, quadratic, or other complexity.
This document provides an overview of various mathematical topics required for an algorithms and data structures course, including:
- Justification for using quantitative methods like mathematics to evaluate algorithms and data structures.
- Common functions like floor, ceiling, and logarithms.
- Properties and proofs of arithmetic series, geometric series, and other polynomial series.
- Introduction of concepts like mathematical induction that will be used throughout the course.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
This document provides a summary of supervised learning techniques including linear regression, logistic regression, support vector machines, naive Bayes classification, and decision trees. It defines key concepts such as hypothesis, loss functions, cost functions, and gradient descent. It also covers generative models like Gaussian discriminant analysis, and ensemble methods such as random forests and boosting. Finally, it discusses learning theory concepts such as the VC dimension, PAC learning, and generalization error bounds.
The document summarizes numerical integration methods for solving equations of motion directly in the time domain, including explicit and implicit methods. It describes Newmark's β method, the central difference method, and Wilson-θ method. Key steps involve discretizing the equations of motion and relating response parameters at different time steps using finite difference approximations. Stability, accuracy, and error considerations are also discussed.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter (Lambda), which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter Lambda, which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
I am Kennedy L. I am a Proficient Computer Network Assignment Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, the University of Sydney, Australia. I have been helping students with their assignments for the past 7 years. I solve assignments related to Proficient Computer Network.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Proficient Computer Network Assignment.
This document presents a modified q-homotopy analysis method (mq-HAM) for solving high-order nonlinear partial differential equations. The mq-HAM improves upon the standard q-HAM by avoiding repeated computations at each iteration step, making it more efficient. As an illustrative example, the method is applied to solve second- and third-order nonlinear cases. The key steps of the mq-HAM include transforming the high-order PDE into a system of first-order equations, then obtaining a series solution through a zero-order deformation equation and its derivatives with respect to an embedded parameter.
This document discusses algorithms and their analysis. It begins by defining an algorithm as a precise set of instructions to solve a problem or perform a computation. Key properties of algorithms are described, including having specified inputs/outputs and being finite, definite, and effective. Examples are given of algorithms to find the maximum element, perform linear search, and perform binary search on an ordered list. The document then discusses analyzing the time and space complexity of algorithms as their input size increases. Common complexity functions like constant, logarithmic, linear, quadratic, and exponential time are described. Rules for analyzing the complexity of combined operations are provided. Finally, examples are worked through to determine the time complexity of algorithms that find the maximum difference between elements in a list
Efficient end-to-end learning for quantizable representationsNAVER Engineering
발표자: 정연우(서울대 박사과정)
발표일: 2018.7.
유사한 이미지 검색을 위해 neural network를 이용해 이미지의 embedding을 학습시킨다. 기존 연구에서는 검색 속도 증가를 위해 binary code의 hamming distance를 활용하지만 여전히 전체 데이터 셋을 검색해야 하며 정확도가 떨어지는 다는 단점이 있다. 이 논문에서는 sparse한 binary code를 학습하여 검색 정확도가 떨어지지 않으면서 검색 속도도 향상시키는 해쉬 테이블을 생성한다. 또한 mini-batch 상에서 optimal한 sparse binary code를 minimum cost flow problem을 통해 찾을 수 있음을 보였다. 우리의 방법은 Cifar-100과 ImageNet에서 precision@k, NMI에서 최고의 검색 정확도를 보였으며 각각 98× 와 478×의 검색 속도 증가가 있었다.
Linear block codes add redundancy to data by encoding it into blocks of n coded bits from k information bits, forming (n,k) block codes. The encoder uses a generator matrix to map k message bits to n codeword bits. Codewords can be systematically encoded by placing the k message bits first followed by (n-k) parity check bits. The parity check matrix H defines the parity check equations to detect errors in the received codeword. The syndrome of a received word indicates the error pattern. The minimum distance of a block code is the smallest Hamming distance between distinct codewords and determines its error correction capability. Linear block codes are widely used in communications and storage for their simplicity in implementing error detection and correction.
This document discusses automatic Bayesian cubature for numerical integration. It begins with an introduction to multivariate integration and the challenges it poses. It then describes an automatic cubature algorithm that generates sample points and computes error bounds iteratively until a tolerance threshold is met. Next, it covers Bayesian cubature, which treats integrands as random functions to obtain probabilistic error bounds. It defines a Bayesian trio identity relating the integration error to discrepancies, variations, and alignments. The document concludes with discussions of future work.
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
This summary provides the key details from the document in 3 sentences:
The document presents a new iterative method (M2 method) for determining the exact solution to a parametric linear programming problem where the objective function and constraints contain parameters. The M2 method exploits the concept of a p-solution to a square linear interval parametric system and iteratively reduces the parameter domain while maintaining upper and lower bounds on the optimal objective value. A numerical example is given to illustrate the new iterative approach for solving parametric linear programming problems.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
This document discusses methods for performing sparse time-frequency representation (STFR) on signals in 2 dimensions. STFR decomposes signals into intrinsic mode functions (IMFs) by finding the sparsest representation of the signal over a redundant dictionary. The 1D version has been implemented successfully, but extending to 2D is challenging due to the need to update in two directions simultaneously. Several attempted algorithms are described, including applying the 1D algorithm to slices of the 2D signal in different directions and averaging the results. The most recent attempt uses bi-directional slicing to overcome issues with previous global approaches.
Introducción al Análisis y diseño de algoritmosluzenith_g
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms to determine their time and space complexity, and how this involves determining how the resources required grow with the size of the problem. It provides examples of analyzing simple algorithms and determining whether they have linear, quadratic, or other complexity.
This document provides an overview of various mathematical topics required for an algorithms and data structures course, including:
- Justification for using quantitative methods like mathematics to evaluate algorithms and data structures.
- Common functions like floor, ceiling, and logarithms.
- Properties and proofs of arithmetic series, geometric series, and other polynomial series.
- Introduction of concepts like mathematical induction that will be used throughout the course.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
This document provides a summary of supervised learning techniques including linear regression, logistic regression, support vector machines, naive Bayes classification, and decision trees. It defines key concepts such as hypothesis, loss functions, cost functions, and gradient descent. It also covers generative models like Gaussian discriminant analysis, and ensemble methods such as random forests and boosting. Finally, it discusses learning theory concepts such as the VC dimension, PAC learning, and generalization error bounds.
The document summarizes numerical integration methods for solving equations of motion directly in the time domain, including explicit and implicit methods. It describes Newmark's β method, the central difference method, and Wilson-θ method. Key steps involve discretizing the equations of motion and relating response parameters at different time steps using finite difference approximations. Stability, accuracy, and error considerations are also discussed.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter (Lambda), which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
A generalized class of normalized distance functions called Q-Metrics is described in this presentation. The Q-Metrics approach relies on a unique functional, using a single bounded parameter Lambda, which characterizes the conventional distance functions in a normalized per-unit metric space. In addition to this coverage property, a distinguishing and extremely attractive characteristic of the Q-Metric function is its low computational complexity. Q-Metrics satisfy the standard metric axioms. Novel networks for classification and regression tasks are defined and constructed using Q-Metrics. These new networks are shown to outperform conventional feed forward back propagation networks with the same size when tested on real data sets.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
I am Kennedy L. I am a Proficient Computer Network Assignment Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, the University of Sydney, Australia. I have been helping students with their assignments for the past 7 years. I solve assignments related to Proficient Computer Network.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Proficient Computer Network Assignment.
This document presents a modified q-homotopy analysis method (mq-HAM) for solving high-order nonlinear partial differential equations. The mq-HAM improves upon the standard q-HAM by avoiding repeated computations at each iteration step, making it more efficient. As an illustrative example, the method is applied to solve second- and third-order nonlinear cases. The key steps of the mq-HAM include transforming the high-order PDE into a system of first-order equations, then obtaining a series solution through a zero-order deformation equation and its derivatives with respect to an embedded parameter.
This document discusses algorithms and their analysis. It begins by defining an algorithm as a precise set of instructions to solve a problem or perform a computation. Key properties of algorithms are described, including having specified inputs/outputs and being finite, definite, and effective. Examples are given of algorithms to find the maximum element, perform linear search, and perform binary search on an ordered list. The document then discusses analyzing the time and space complexity of algorithms as their input size increases. Common complexity functions like constant, logarithmic, linear, quadratic, and exponential time are described. Rules for analyzing the complexity of combined operations are provided. Finally, examples are worked through to determine the time complexity of algorithms that find the maximum difference between elements in a list
Efficient end-to-end learning for quantizable representationsNAVER Engineering
발표자: 정연우(서울대 박사과정)
발표일: 2018.7.
유사한 이미지 검색을 위해 neural network를 이용해 이미지의 embedding을 학습시킨다. 기존 연구에서는 검색 속도 증가를 위해 binary code의 hamming distance를 활용하지만 여전히 전체 데이터 셋을 검색해야 하며 정확도가 떨어지는 다는 단점이 있다. 이 논문에서는 sparse한 binary code를 학습하여 검색 정확도가 떨어지지 않으면서 검색 속도도 향상시키는 해쉬 테이블을 생성한다. 또한 mini-batch 상에서 optimal한 sparse binary code를 minimum cost flow problem을 통해 찾을 수 있음을 보였다. 우리의 방법은 Cifar-100과 ImageNet에서 precision@k, NMI에서 최고의 검색 정확도를 보였으며 각각 98× 와 478×의 검색 속도 증가가 있었다.
Linear block codes add redundancy to data by encoding it into blocks of n coded bits from k information bits, forming (n,k) block codes. The encoder uses a generator matrix to map k message bits to n codeword bits. Codewords can be systematically encoded by placing the k message bits first followed by (n-k) parity check bits. The parity check matrix H defines the parity check equations to detect errors in the received codeword. The syndrome of a received word indicates the error pattern. The minimum distance of a block code is the smallest Hamming distance between distinct codewords and determines its error correction capability. Linear block codes are widely used in communications and storage for their simplicity in implementing error detection and correction.
This document discusses automatic Bayesian cubature for numerical integration. It begins with an introduction to multivariate integration and the challenges it poses. It then describes an automatic cubature algorithm that generates sample points and computes error bounds iteratively until a tolerance threshold is met. Next, it covers Bayesian cubature, which treats integrands as random functions to obtain probabilistic error bounds. It defines a Bayesian trio identity relating the integration error to discrepancies, variations, and alignments. The document concludes with discussions of future work.
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
This summary provides the key details from the document in 3 sentences:
The document presents a new iterative method (M2 method) for determining the exact solution to a parametric linear programming problem where the objective function and constraints contain parameters. The M2 method exploits the concept of a p-solution to a square linear interval parametric system and iteratively reduces the parameter domain while maintaining upper and lower bounds on the optimal objective value. A numerical example is given to illustrate the new iterative approach for solving parametric linear programming problems.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Design and optimization of ion propulsion dronebjmsejournal
Electric propulsion technology is widely used in many kinds of vehicles in recent years, and aircrafts are no exception. Technically, UAVs are electrically propelled but tend to produce a significant amount of noise and vibrations. Ion propulsion technology for drones is a potential solution to this problem. Ion propulsion technology is proven to be feasible in the earth’s atmosphere. The study presented in this article shows the design of EHD thrusters and power supply for ion propulsion drones along with performance optimization of high-voltage power supply for endurance in earth’s atmosphere.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Rainfall intensity duration frequency curve statistical analysis and modeling...bijceesjournal
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period (1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal, normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall, 92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation, wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in making appropriate decisions in managing and minimizing floods in the study area.
4. Introduction
Fast convolution:
Implementation of convolution algorithm using fewer multiplication operations by
algorithmic strength reduction.
Algorithmic Strength Reduction: Reducing the number of strong operations (such as
multiplication operations) at the expense of an increase in the number of weak
operations (such as addition operations).
Reduce computation complexity.
Best suited for implementation using either programmable or dedicated hardware.
5. Introduction
Fast convolution:
Example: Consider the complex number multiplication: (a+jb)(c+dj)=e+jf, where {a, b, c,
d, e, f } ∈ R. (R: set of real numbers).
The direct implementation requires
four multiplications and two additions:
The number of multiplications can be reduced to 3 at the expense of 3 extra additions
by using the identities:
Three Multiplications
Five Additions
6. Introduction
Fast convolution:
Example: (contd…)
In matrix form, its coefficient matrix can be decomposed as the product of a 2X3(C), a
3X3(H)and a 3X2(D) matrix:
Where C is a post-addition matrix (requires two additions), D is a pre-addition matrix
(requires one addition), and H is a diagonal matrix (requires two additions to get its
diagonal elements).
7. Introduction
Fast convolution:
Example: (contd…)
The arithmetic complexity is reduced to three multiplications and three additions (not
including the additions in H matrix).
Two well-known approaches for the design of fast short-length convolution algorithms:
1. Cook-Toom algorithm (based on Lagrange Interpolation) and
2. Winograd Algorithm (based on the Chinese remainder theorem)
8. A linear convolution algorithm for polynomial multiplication based on the
Lagrange Interpolation Theorem.
Lagrange Interpolation Theorem:
Let β0, β1, ...., βn be a set of (n +1) distinct points, and let f(βi) , for i = 0, 1, …, n be
given. There is exactly one polynomial f ( p) of degree “n” or less that has value f(βi)
when evaluated at βi for i = 0, 1, …, n.
It is given by:
Cook-Toom algorithm:
9. Lagrange Interpolation Theorem: (contd…)
Interpolation is a special case of data fitting, in which the goal is to find a linear
combination of “n” known functions to fit a set of data that imposes “n” constraints,
thus guaranteeing a unique solution that fits the data exactly, rather than
approximately.
Data fitting: Constructing a continuous function from a set of discrete data points,
where the data is obtained from measurements.
Cook-Toom algorithm: (contd…)
10. Cook-Toom algorithm: (contd…)
Lagrange Interpolation Theorem: (contd…)
Compute the unique polynomial pn(x) of degree “n” that satisfies pn(xi) = yi, i = 0, . . . , n,
where the points (xi, yi) are given.
The points x0, x1, . . . , xn are called interpolation points.
The polynomial pn(x) is called the interpolating polynomial of the data (x0, y0), (x1, y1),
. . ., (xn, yn).
Form the system Ax = b, where bi = yi , i = 0, . . . , n for computing the interpolation
polynomial
11. Lagrange Interpolation Theorem: (contd…)
The entries of A are defined by aij = pj (xi); i, j = 0, . . . , n, where x0, x1, . . . , xn are the
points at which the data y0, y1, . . . , yn are obtained, and pj (x) = xj , j = 0, 1, . . . , n.
The basis {1, x, . . . , xn} of the space of polynomials of degree (n+1) is called the
monomial basis.
The corresponding matrix A is called the Vandermonde matrix for the points x0, x1, . . . ,
xn.
Cook-Toom algorithm: (contd…)
12. Lagrange Interpolation Theorem: (contd…)
In Lagrange interpolation, the matrix A is simply the identity matrix.
The interpolating polynomial is written as:
Where the polynomials have the property that:
Cook-Toom algorithm: (contd…)
13. Lagrange Interpolation Theorem: (contd…)
The polynomials {£n,j}, j = 0, … , n are called the Lagrange polynomials for the
interpolation points x0, x1, … , xn.
They are defined by:
Cook-Toom algorithm: (contd…)
14. Convolution basics:
Given two sequences:
o Data sequence di, 0 ≤ i≤ N-1, of length N.
o Filter sequence gi, 0 ≤ i≤ L-1, of length L
The Linear convolution these two sequences is given by:
(NL multiplications)
Cook-Toom algorithm: (contd…)
15. Convolution basics: (contd…)
Express the convolution in the notation of polynomials:
o Let and , then
where
Cook-Toom algorithm: (contd…)
16. An algorithm for linear convolution by using multiplying polynomials.
Consider the following system:
The application of Lagrange interpolation theorem into linear convolution.
Consider an N-point sequence h = {h0, h1, ... , hN-1} and an L-point sequence x = {x0, x1, ...
, xL-1}.
Cook-Toom algorithm: (contd…)
17. The linear convolution of h and x can be expressed in terms of polynomial
multiplication as follows: s( p) = h( p) . x( p), where
The output polynomial s(p) has degree (L+N-2) and has (L+N-1) different points.
Cook-Toom algorithm: (contd…)
18. s(p) can be uniquely determined by its values at (L+N-1) different points.
Let {β0, β1, ... , βL+N-2} be (L+N-1) different real numbers.
If s(βi) for i = {0,1, ... , (L+N-2)} are known, then s(p) can be computed using the
Lagrange interpolation theorem as:
This equation is the unique solution to compute linear convolution for s(p) given the
values of s(βi ) , for i = {0,1, ..., (L+N-2)}.
Cook-Toom algorithm: (contd…)
19. Algorithm Description:
1. Choose (L+N-1) different real numbers β0, β1, … , βL+N -2.
2. Compute h(βi ) and x(βi ), for i = {0,1, … , (L+N-2)}.
3. Compute s(βi) = h(βi) . x(βi), for i = {0,1, … , (L+N-2)}.
4. Compute s(p) by using:
Cook-Toom algorithm: (contd…)
20. Algorithm Complexity:
The goal of the fast-convolution algorithm is to reduce the multiplication complexity.
So, if βi ‘s (where i=0, 1, …, L+N-2) are chosen properly, the computation in step-2
involves some additions and multiplications by small constants.
The multiplications are only used in step-3 to compute s(βi).
So, only (L+N-1) multiplications are needed.
Cook-Toom algorithm: (contd…)
21. Advantages :
1. The number of multiplications is reduced from (LN) to (L+N-1) at the expense of an
increase in the number of additions.
2. An adder has much less area and computation time than a multiplier.
3. Lead to large savings in hardware (VLSI) complexity and generate computationally
efficient implementation.
Cook-Toom algorithm: (contd…)
22. Example: Construct a (2X2) convolution algorithm using Cook-Toom
algorithm with β={0,1,-1}.
Solution:
Write the (2X2) convolution in polynomial multiplication form as:
s (p) = h (p) . x (p), where h (p) = h0 + h1p . x (p)
i.e., s (p) = s0 + s1p + s2p2
Direct implementation requires four multiplications and one addition.
can be expressed in matrix form as follows:
Cook-Toom algorithm: (contd…)
23. Example: Construct a (2X2) convolution algorithm using Cook-Toom
algorithm with β={0,1,-1}.
Solution: (contd…)
The direct implementation can be expressed in matrix form as follows:
Cook-Toom algorithm: (contd…)
24. Example: Construct a (2X2) convolution algorithm using Cook-Toom
algorithm with β={0,1,-1}.
Solution: (contd…)
Use Cook-Toom algorithm to get an efficient convolution implementation with reduced
multiplication number:
1. Step 1: Choose β0=0, β1=1, β2=-1
2. Step 2 :
Cook-Toom algorithm: (contd…)
25. Example: Construct a (2X2) convolution algorithm using Cook-Toom
algorithm with β={0,1,-1}.
Solution: (contd…)
3. Step 3 : Calculate s(β0), s(β1), s(β2) by using 3 multiplications.
Cook-Toom algorithm: (contd…)
28. Solution: (contd…)
The computation is carried out as follows (5 additions, 3 multiplications):
Hence the Cook-Toom algorithm needs three multiplications and five additions
(ignoring the additions in the pre-computation ), i.e., the number of multiplications is
reduced by one at the expense of four extra additions.
Cook-Toom algorithm: (contd…)
29. Conclusions:
Some additions in the pre-addition or post-addition matrices can be shared.
So, when we count the number of additions, we count only one, instead of two or three.
If we take h0, h1 as the FIR filter coefficients and take x0, x1 as the signal (data)
sequence, then the terms H0, H1 need not be recomputed each time the filter is used.
They can be precomputed once offline and stored. (So, we ignore these computations
when counting the number of operations).
Cook-Toom algorithm: (contd…)
30. Conclusions:
The Cook-Toom algorithm is a matrix decomposition.
A convolution can be expressed in matrix-vector forms as:
Generally, the equation can be expressed as:
Where C is the post-addition matrix, D is the pre-addition matrix, and H is a diagonal
matrix with Hi, i = 0, 1, …, (L+N-2) on the main diagonal.
Cook-Toom algorithm: (contd…)
31. Conclusions:
Since T=CHD, it implies that the Cook-Toom algorithm provides a way to factorize the
convolution matrix T into multiplication of 1 post-addition matrix C, 1 diagonal
matrix H and 1 pre-addition matrix D, such that the total number of multiplications is
determined only by the non-zero elements on the main diagonal of the diagonal
matrix H (note matrices C and D contain only small integers).
Limitations:
Although the number of multiplications is reduced, the number of additions has
increased.
Cook-Toom algorithm: (contd…)
32. The Cook-Toom algorithm is used to further reduce the number of addition
operations in linear convolutions.
Define s' (p) = s(p) - SL+N-2 . pL+N-2.
Notice that the degree of s(p) is L+N-2 and SL+N-2 is its highest order coefficient.
Therefore the degree of s' (p) is L+N-3.
This modified polynomial s’ (p) is used in modified Cook-Toom algorithm.
Modified Cook-Toom algorithm:
33. Algorithm Description:
1. Choose L+N-2 different real numbers β0, β1, … , βL+N-3
2. Compute h(βi) and x(βi) , for i ={0,1, … , L+N-3}
3. Compute s(βi) = h(βi) . x(βi), for i ={0,1, … , L+N-3}
4. Compute s'(βi) = s(βi)-(sL+N-2).(βi)L+N-2, for i ={0,1, … , L+N-3}
Modified Cook-Toom algorithm: (contd…)
35. Example: Derive a (2X2) convolution algorithm using the modified
Cook-Toom algorithm with β={0,-1}.
Solution:
Consider the Lagrange interpolation for s'(p) = s(p) - h1x1p2 at {β0 = 0, β1 = -1}
Find, s'(βi) = h(βi)x(βi) - h1x1βi
2
Find,
Modified Cook-Toom algorithm: (contd…)
36. Solution: (contd…)
Find,
This computation requires 2 multiplications {one for h0x0 and another for (h0-h1)(x0-
x1); not counting that for h1x1 multiplication}.
Apply the Lagrange interpolation
algorithm, we get:
Modified Cook-Toom algorithm: (contd…)
39. Solution: (contd…)
The computation is carried out as follows:
The total number of operations are 3 multiplications (three in step-3) and 3 additions
(one in step-2 and two in step-4).
Compared with the linear convolution using Cook-Toom algorithm, the number of
addition operations has been reduced by 2 while the number of multiplications
remains the same
Modified Cook-Toom algorithm: (contd…)
40. Conclusions:
Limitations:
1. Although the number of multiplications is reduced, the number of additions has
increased (As the size of the problem increases, the number of additions increase
rapidly).
2. The choices of βi=0, ±1 are good, while the choices of ±2, ±4 (or other small integers)
result in complicated pre-addition and post-addition matrices.
3. For larger problems, Cook-Toom algorithm becomes cumbersome.
Cook-Toom algorithms:
41. The Winograd short convolution algorithm:
Based on the CRT (Chinese Remainder Theorem).
Chinese Remainder Theorem:
It’s possible to uniquely determine a nonnegative integer given its remainder with respect
to the given moduli, provided that the moduli are relatively prime and the integer is
known to be smaller than the product of the moduli.
Winograd Algorithm:
42. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers:
Review of Integer Ring:
For every integer c and positive integer d, there is a unique pair of integer Q, called the
quotient, and integer s, called the remainder; such that c=dQ + s, where 0 ≤ s ≤ d-1.
i.e., Q= ⎣ c/d ⎦ and s=Rd[c]
Euclidean Algorithm: Given two positive integers s and t, where (t < s), then their
Greatest Common Divisor (GCD) can be computed by an iterative application of the
division algorithm.
Winograd Algorithm: (contd…)
43. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Euclidean Algorithm:
1. The process stops when a remainder of zero is
obtained.
2. The last nonzero remainder t(n) is the GCD(s,t).
Winograd Algorithm: (contd…)
44. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Euclidean Algorithm:
In matrix notation:
For any integer s and t, there exists integers a and b such that GCD[s, t]= as + bt
It is possible to uniquely determine a nonnegative integer given its moduli with
respect to each of several integers, provided that the integer is known to be smaller
than the product of the moduli.
Winograd Algorithm: (contd…)
45. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Euclidean Algorithm:
Example: {m1=3, m2=5} and M=3×5=15
Winograd Algorithm: (contd…)
46. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Euclidean Algorithm: Example: {m1, m2, m3} = {3, 4, 5} and M = 3× 4 × 5 = 60
Winograd Algorithm: (contd…)
47. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Given a set of integers m0, m1, … , mk that are pairwise relatively prime (co-prime),
then for each integer c, 0 ≤ c < M= m0m1…mk, there is a one-to-one map between c and
the vector of residues: (Rm0 [c], Rm1[c], … , Rmk [c])
Conversely, given a set of co-prime integers m0, m1, …, mk and a set of integers c0, c1, …,
ck with ci < mi . Then the system of equations ci =c (mod mi), i=0,1,…,k has at most one
solution for 0 ≤ c < M.
Winograd Algorithm: (contd…)
48. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Define Mi=M/mi, then GCD[Mi, mi]=1. So there exists integers Ni and ni with GCD[Mi,
mi]= 1 = NiMi + nimi, i=0,1,…,k.
The system of equations ci=c (mod mi), 0 ≤ I ≤ k, is uniquely solved by:
Winograd Algorithm: (contd…)
49. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Example: GCD
Winograd Algorithm: (contd…)
50. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
Given ci = Rmi[c] (represents the remainder when cis divided by mi), for i = 0,1,...,k ,
where miare moduli and are relatively prime, then c = 𝑖=0
𝑘
𝑐𝑖 NiMi mod M, where
M = 𝑖=0
𝑘
𝑚𝑖, Mi = M/mi, and Niis the solution of NiMi + nimi = GCD(Mi, mi)=1,
provided that 0 ≤ c < M
Winograd Algorithm: (contd…)
51. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Integers: (contd…)
If NiMi + nimi = GCD (Mi, mi) = 1, then we have NiMi= 1 (mod mi).
C (mod mi) = 𝑖=0
𝑘
𝑐𝑖𝑁𝑖𝑀𝑖 (𝑚𝑜𝑑 𝑚𝑖)
= ci NiMi (mod mi)
= ci(mod mi)
Winograd Algorithm: (contd…)
52. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem (CRT) for Integers: (contd…)
Example: Using the CRT for integer, Choose moduli m0=3, m1=4, m2=5.
Then M = m0m1m2 = 60 and M i = M/mi
By Euclidean theorem, we have:
Where and are obtained using the Euclidean GCD algorithm.
Given that the integer c satisfy: 0 ≤ c < M, Let ci = Rmi[c]
Winograd Algorithm: (contd…)
53. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem (CRT) for Integers: (contd…)
Example:
The integer c can be calculated as:
For c=17:
Winograd Algorithm: (contd…)
54. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem (CRT) for Integers: (contd…)
By taking residues, large integers are broken down into small pieces (that may be
easy to add and multiply).
Examples:
Winograd Algorithm: (contd…)
55. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Polynomials:
Winograd Algorithm: (contd…)
56. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Polynomials: (contd…)
Given a set of polynomials m(0)(x), m(1)(x), …, m(k)(x), that are pair-wise relatively
prime (co-prime), then for each polynomial c(x), deg(c(x)) < deg(M(x)), M(x)=
m(0)(x)m(1)(x)…m(k)(x), there is a one-to-one map between c(x) and the vector of
residues:
Winograd Algorithm: (contd…)
57. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Polynomials: (contd…)
Conversely, given a set of co-prime polynomials m(0)(x), m(1)(x), …, m(k)(x) and a
set of polynomials c(0)(x), c(1)(x), …, c(k)(x) with deg(c(i)(x)) < deg(m(i)(x)).
Then the system of equations c(i)(x) =c(x) (mod m(i)(x)), i=0,1,…,k has at most
one solution for deg(c(x)) < deg(M(x)).
Winograd Algorithm: (contd…)
58. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Polynomials: (contd…)
Define M(i)(x)=M(x)/m(i)(x), then GCD[M(i)(x),m(i)(x)]=1.
So there exists polynomials N(i)(x) and n(i)(x) with GCD[M(i)(x), m(i)(x)] = 1 =
N(i)(x)M(i)(x) + n(i)(x)m(i)(x), i=0,1,…,k.
The system of equations c(i)(x) =c(x) (mod m(i)(x) ), 0 ≤ I ≤ k, is uniquely solved
by:
Winograd Algorithm: (contd…)
59. The Winograd short convolution algorithm: ( contd…)
Chinese Remainder Theorem for Polynomials: (contd…)
The remainder of a polynomial with regard to modulus xi + f(x), where deg(f(x)) <
i, can be evaluated. by substituting xi by –f(x) in the polynomial
Example:
Winograd Algorithm: (contd…)
60. The Winograd short convolution algorithm: ( contd…)
Algorithm Description:
1. Choose a polynomial with degree higher than the degree of h(p) x(p ) and factor it
into (k+1) relatively prime polynomials with real coefficients,
i.e., m(p)= m(0)(p)m(1)(p) … m(k )(p).
2. Let M(i)( p) = m(p)/m(i) (p). Use the Euclidean GCD algorithm to solve N(i)(p)M(i)(p) +
n(i)(p)m(i)(p) = 1 for N(i)(p)
Winograd Algorithm: (contd…)
61. The Winograd short convolution algorithm: ( contd…)
Algorithm Description: (contd…)
3. Compute: h(i)(p) = h(p) mod m(i)(p) and x(i)(p) = x(p) mod m(i)(p) for i= 0, 1,
…, k.
4. Compute: S(i)(p) = h(i)(p) x(i)(P) mod m(i)(p) for i =0, 1, …, k.
5. Compute s(p) by using:
Winograd Algorithm: (contd…)
62. The Winograd short convolution algorithm: ( contd…)
Example: Consider a (2X3) linear convolution S(p) = h(p)x(p) where h(p) = h0 +
h1(p) and x(p) = x0 + x1p + x2p2. Construct an efficient realization using Winograd
algorithm with m(p) = p(p-1)(p2+1).
Solution:
Let m(0)(p) = p, m(1)(p) = (p-1) and m(2)(p) = (p2+1).
Construct the following table using the relationships: M(i)(p) = m(p)/m(i)(p) and
N(i)(p)M(i)(p) + n(i)(p)m(i)(p) = 1 for i =0, 1, 2.
Winograd Algorithm: (contd…)
63. The Winograd short convolution algorithm: ( contd…)
Solution: (contd…)
Compute residues from h(p) = h0 + h1 p and x(p) = x0 + x1p + x2p2:
i.e., h(0)(p) = h0 and x(0)(p) = x0.
h(1)(p) = h0 + h1 and x(1)(p) = x0 + x1 + x2.
h(2)(p) = h0 + h1p and x(2)(p) = (x0 – x2) + x1p.
Winograd Algorithm: (contd…)
No multiplication
operations required
64. The Winograd short convolution algorithm: ( contd…)
Solution: (contd…)
s(0)(p) = h0x0 = s0
(0)
s(1)(p) = (h0 + h1)(x0 + x1 + x2) = s0
(1)
s(2)(p) = (h0 + h1p)((x0-x2)+x1p) mod(p2+1)
= h0(x0-x2)-h1x1+(h0x1+h1(x0-x2))p
= s0(2) + s1
(2)p
Algorithm complexity: One multiplication for s(0)(p), one for s(1)(p) and four for s(2)(p)
Winograd Algorithm: (contd…)
65. The Winograd short convolution algorithm: ( contd…)
Solution: (contd…)
The computational complexity can be reduced to three multiplications as shown
below:
Then:
Winograd Algorithm: (contd…)
66. The Winograd short convolution algorithm: ( contd…)
Solution: (contd…)
Substitute s(0)(p), s(1)(p) and s(2)(p) in the relation for s(p) to obtain the following
table:
Winograd Algorithm: (contd…)
67. The Winograd short convolution algorithm: ( contd…)
Solution: (contd…)
Therefore, we have:
Notice that:
Winograd Algorithm: (contd…)
68. The Winograd short convolution algorithm: ( contd…)
Solution: (contd…)
Finally, we have:
Here the Winograd convolution algorithm requires five multiplications and 11
additions compared with six multiplications and two additions for direct
implementation.
Winograd Algorithm: (contd…)
69. The Winograd short convolution algorithm: ( contd…)
Conclusions:
The number of multiplications in Winograd algorithm is highly dependent
on the degree of each m(i )( p).
Therefore, the degree of m(p) should be as small as possible.
More efficient form (or a modified version) of the Winograd algorithm can be
obtained by letting deg[m(p)]= deg[s(p)] and applying the CRT to:
s’(p) = s(p)-hN-1 . xL-1.m(p)
Winograd Algorithm: (contd…)
70. Known as circular convolution.
Let the filter coefficients be h = {h0, h1, … , hn-1}, and the data sequence be x ={x0,
x1, … , xn-1}.
The cyclic convolution can be expressed as:
The output samples are given by:
where ((i-k)) denotes (i-k) mod n
The cyclic convolution can be computed as a linear convolution reduced by modulo
pn – 1 (Notice that there are 2n-1 different output samples for this linear convolution)
Cyclic Convolution:
71. In circular convolution, coefficients with indices larger than (n-1) are folded back into
terms with indices small than n.
We can express the cyclic convolution by polynomial product :
The circular convolution system is shown below:
Cyclic Convolution: (contd…)
72. The direct computation of cyclic convolution:
Cyclic Convolution: (contd…)
73. Alternatively, the cyclic convolution can be computed using CRT with
m( p) = pn -1, which is much simpler.
Example: Construct a (4X4) cyclic convolution algorithm using CRT with m( p) = p4-1
= (p - 1)(p + 1)(p2 + 1)
Solution:
Let h( p) = h0 + h1 p + h2 p2 + h p3 and x( p) = x0 + x1 p + x2 p2 + x3 p3
Let m(0 )( p) = p -1, m (1)( p) = p +1 and m(2 )( p) = p2 +1.
Cyclic Convolution: (contd…)
74. Solution: (contd…)
Get the following table using the relationships M (i)(p) = m(p)/m(i)(p) and
N (i)(p)M (i)(p) + n (i)(p)m(i)(p) = 1.
Cyclic Convolution: (contd…)
79. Solution: (contd…)
Therefore, we have:
This algorithm requires five multiplications and 15 additions.
The direct implementation requires 16 multiplications and 12 additions.
An efficient cyclic convolution algorithm can often be easily extended to construct
efficient linear convolution.
Cyclic Convolution: (contd…)
80. Solution: (contd…)
The direct implementation requires 16 multiplications and 12 additions as shown in
the following matrix-form:
Notice that the cyclic convolution matrix is a circulant matrix.
Cyclic Convolution: (contd…)